Simplify the following expression and state the condition under which the simplification is valid: $y = \dfrac{a^2 + 6a}{a^2 - a - 42}$
Explanation: First factor the expressions in the numerator and denominator. $ \dfrac{a^2 + 6a}{a^2 - a - 42} = \dfrac{(a)(a + 6)}{(a - 7)(a + 6)} $ Notice that the term $(a + 6)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(a + 6)$ gives: $y = \dfrac{a}{a - 7}$ Since we divided by $(a + 6)$, $a \neq -6$. $y = \dfrac{a}{a - 7}; \space a \neq -6$